UBC Theses and Dissertations
Application of chemotactic models to Alzheimer’s disease Luca, Magdalena
This thesis presents the analysis of several chemotactic models developed for pattern formation in biological systems. The project was motivated by the complex interactions between glial cells, cytokines and proteins that cause neuronal death and eventual dementia in Alzheimer's disease. I investigate conditions that lead to periodic patterns and aggregation of glial cells and chemicals, as observed in the senile plaques of Alzheimer's disease. I have examined a hierarchy of one dimensional models, starting with a model in which there is a single chemoattractant inflammatory agent and the microglia cells. Thus, I constructed a minimal partial differential equations model similar to the Keller-Segel model for the chemotaxis of microglia and their involvement in secretion and uptake of various inflammatory factors. Stationary solutions of the system of equations are investigated using a method developed by Schaaf. The linear stability analysis of the system leads to the aggregation condition which can be biologically interpreted. As a result, I suggest possible methods that might affect plaque formation in Alzheimer's disease. Numerical simulations help corroborate theoretical results. A different form of the kinetics function which is part of the Keller-Segel model is suggested. Thus, the limiting kinetics function in the chemoattractant equation incorporates production or removal of the chemical factor by the cells, depending on the model parameters. Also, it limits the sharp increase in the cell density which is well known to occur in this type of a chemotactic model. Finally, a model for chemotaxis of cells in response to a chemoattractant and a chemorepellent is developed to include a greater level of detail. The system is reduced in complexity using quasi-steady state approximations, and explicit solutions for chemical diffusion are found using the Green's function method. The model predicts that local attraction and long range repulsion create periodic pattern formation. Biological interpretations of the linear instability condition lead to suggestions for therapy interventions in Alzheimer's disease. Using real biological parameters available in the literature, I compare and discuss the applicability of this particular model to the pattern formation observed in Alzheimer's disease.
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